Algorithm and a Method for Characterizing Fractal Volumes

ABSTRACT

A computer implemented method for obtaining an analytical representation of an internal structure and spatial properties distribution of a selected physical domain includes identifying d-dimensional correspondences of measured spatial properties or field distributions; and applying an inverse algorithm to the d-dimensional spatial properties or field distributions to calculate the Weierstrass-Mandelbrot (W-M) fractal model to thereby determine parameters defining an analytical and continuous Weierstrass-Mandelbrot (W-M) representation.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application61/671,378 filed on Jul. 13, 2012 and incorporated herein by reference.

FIELD OF THE INVENTION

The present invention is directed to a method for obtaining ananalytical representation of an internal structure and spatialproperties distribution of a selected physical domain, and in particularto applying an inverse algorithm to calculate the W-M fractal model andthereby determine parameters defining an analytical and continuousWeierstrass-Mandelbrot (W-M) representation.

BACKGROUND OF THE INVENTION

The need to adequately represent and model the structure and thebehavior of physical or virtual volumes containing physical propertydistributions or physical scalar fields as they are relevant to variousapplications has motivated the utilization of appropriate fractalfunctions that can be used to represent such volumes.

A lower dimension approach that is useful for determining themechanical, thermal, electrical and fluid properties of the static andthe sliding contact between two different conductors of heat andelectric current involves the full identification of the parameters ofthe 2D Weirstrass-Mandelbrot (W-M) function, described in (Michopoulos,J. G., and Iliopoulos, A. P., 2011. “Complete High Dimensional InverseCharacterization of Fractal Surfaces”, 2011 ASME International DesignEngineering Technical Conferences & Computers and Information inEngineering Conference, Vol. CD-ROM, ASME) (“Michopoulos et al.”) and inU.S. patent application Ser. No. 13/507,968 filed Aug. 8, 2012 andincorporated herein by reference. The present invention represents ageneralization of this methodology for higher dimensional fractalvolumes.

BRIEF SUMMARY OF THE INVENTION

According to the invention, a method for obtaining an analyticalrepresentation of an internal structure and spatial propertiesdistribution of a selected physical domain includes identifyingd-dimensional correspondences of measured spatial properties or fielddistributions; and applying an inverse algorithm to the d-dimensionalspatial properties or field distributions to calculate the W-M fractalmodel to thereby determine parameters defining an analytical andcontinuous fractal Weierstrass-Mandelbrot (W-M) representation.

Exemplary physical domains for application of the method of theinvention are generally any dense or coarse material where the method isapplied to its internal structure as enclosed in its defined volume andsurface such as a metal alloy, a carbon epoxy composite, a biologicaltissue, and any physical or virtual entity with a varying distributionof a scalar field. The physical domain may also be a man-made structure,like a microprocessor, an aircraft, a ship, an automobile, a building orsimilar technological objects. The physical domain alternatively may bea live organism or part of a live organism, or a digital representationlike a 1 dimensional, 2 dimensional a 3 dimensional or a d-dimensionalimage of a real world or of an artificial object—the image can be of anyshape. The physical domain may be a landform, like a mountain, or a lakeor a part of a landform, a planet, or a part of a planet, a solarsystem, or a part of a solar system, a cluster of solar systems, or apart of a cluster of solar systems, a galaxy, or a part of a galaxy, acluster of galaxies, or a part of a cluster of galaxies.

The invention provides an efficient method for determining an analyticalrepresentation of the internal structure and spatial propertiesdistribution of various materials such as composite, granular, andporous materials that have been loaded under various conditions as wellas any physical field such as electric and/or magnetic fields,temperature, etc. In particular, an example of these materials arecarbon epoxy composites for which is identified the parameters of thed-dimensional W-M fractal function from measured data. The inventionprovides a volume parameterization as a multivariate analyticgeneralization of its uni-variate version.

The invention provides exact and full characterization of d-dimensionalfractal volumes. Applications that require or are facilitated by havingsuch an analytical representation of internal microstructure or spatialdistribution of inhomogeneous material properties are enabled by thisapproach. The invention can be extended to the use of a homogeneous meshfor performing finite element analysis where the properties arerepresented by a W-M function, thereby bypassing the need for geometrictailoring of the mesh such as it explicitly represents the volumetricstructure and replaces it with a generalized W-M fractal function.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graph showing the uniform sampling of a sphere (dots) andfour typical sampling vectors;

FIG. 2 is a a block diagram showing the forward problem perspective andthe inverse problem perspective according to the invention;

FIG. 3 is a pseudo-colored isosurface combined with associated densityplots of synthetic and inversely identified volumes of a W-M fractalaccording to the invention, demonstrating the success of the algorithmin this patent;

FIG. 4 is a graph showing the mean error of inverse identificationthrough singular value decomposition vs parameter γ according to theinvention;

FIG. 5 compares noisy data with respective identified fractal volumesaccording to the invention in order to demonstrate the robustnessagainst noise;

FIG. 6 is a graph showing the result of noise in the error of thefractal inverse identification from synthetic data according to theinvention;

FIG. 7 shows 8 of the 243 micro tomography slices according to theinvention;

FIG. 8 compares original and inversely identified micro-tomographicvolumes according to the invention in order to demonstrate the successof the method on real data; and

FIG. 9 is a graph showing the mean absolute error of the inversion as afunction of γ according to the invention.

DETAILED DESCRIPTION OF THE INVENTION

d-Dimensional Modeling

It has been established that a fractal surface represents a very richanalytical representation that can model surface topographies ofmaterials at small scales. This seems to be true especially because ofthe properties of continuity, non-differentiability and self affinity ofspecific types of fractals, that are also desired properties of surfacetopographies. The surface power spectra obeys a power-law relationshipover a wide range of frequencies, because the surface topographiesresemble a random process. Such a surface can be represented in polarcoordinates (r, θ) by a complex function W as

$\begin{matrix}{{{W( {r,\theta} )} = {{L( \frac{G}{L} )}^{D - 2}\sqrt{\frac{\ln \; \gamma}{M}}{\sum\limits_{m = 1}^{M}{\sum\limits_{n = 0}^{N}{{\gamma^{{({D - 3})}n}( {1 - ^{\frac{2\; \pi}{L}\gamma^{n}r\; {\cos {({\theta - a_{m}})}}}} )}^{\; \varphi_{mn}}}}}}},} & (1)\end{matrix}$

where L is the linear size of the domain under consideration, G is thefractal roughness, D is the fractal dimension, φ_(mn) is a table of(usually) random phases (with φ_(mn)ε[0,2π]), γ is a parameter thatcontrols the density of the frequencies, M is the number of superimposedridges and N is a number that is chosen based on the desired highestsampling frequency. The following equation can be used to identify anappropriate value for N,

N=n _(max) =int[log(L/Lc)/log γ],  (2)

with L_(c) being a cut-off wavelength, typically defined either by thehighest sampling frequency, or by a physical barrier.

It should be noted that the parameter γ must be greater than 1 and itusually takes values in the vicinity of 1.5 because of surface flatnessand frequency distribution density considerations. The latter though hasbeen recently debated and only the requirement γ>1 was considered as avalid assumption.

Although the above mentioned representation is very useful in variousphysical contexts, its generalization to higher dimensions can provide aconvenient representation of fields in those dimensions. Eq. (1) can begeneralized to d-dimensions by the formula

$\begin{matrix}{{{W(r)}=={\sqrt{\frac{\gamma}{M}}{\sum\limits_{m = 1}^{M}{\sum\limits_{n = {- \infty}}^{\infty}{{A_{m}( {1 - ^{\; k_{0}\gamma^{n}{n_{m} \cdot r}}} )}{^{\; \varphi_{mn}}( {k_{0}\gamma^{n}} )}^{D - {({d + 1})}}}}}}},} & (3)\end{matrix}$

where n_(m) are unit vectors spaced uniformly over a unit d-dimensionalhypersphere. k₀ can be substituted by 2 π/L, parameter A can besubstituted by 2π(2 π/G)^(2-D) and n can be bound between 0 and N. Then(3) can be written as:

$\begin{matrix}{{W(r)} = {{L( \frac{G}{L} )}^{D - 2}\sqrt{\frac{\ln \; \gamma}{M}}{\sum\limits_{m = 1}^{M}{\sum\limits_{n = 0}^{N}{{\gamma^{n{({D - {({d + 1})}})}}( {1 - ^{\; \frac{2\pi}{L}\gamma^{n}{n_{m} \cdot r}}} )}{^{\; \varphi_{mn}}.}}}}}} & (4)\end{matrix}$

The construction of the unit vectors n_(m) can be achieved with the helpof the hyperspherical coordinates

$\begin{matrix}{ \begin{matrix}{x_{1} = {r\; \cos \; \theta_{1}}} \\{x_{2} = {r\; \sin \; \theta_{1}\cos \; \theta_{2}}} \\{x_{3} = {r\; \sin \; \theta_{1}\sin \; \theta_{2}\cos \; \theta_{3}}} \\\vdots \\{x_{d} = {r\; \sin \; \theta_{1}\sin \; \theta_{2}\mspace{14mu} \ldots \mspace{14mu} \sin \; \theta_{d - 2}\sin \; \theta_{d - 1}}}\end{matrix} \},} & (5)\end{matrix}$

where θ_(i) are the angular coordinates. For unit vectors, one can user=1. It should be noted that in order to avoid traversing thehypersphere twice we can choose to traverse θ₁ . . . θ_(d-2) over therange [0,π]. A uniform sampling of the hypersphere can be achieved bysetting,

$\begin{matrix}{ \begin{matrix}{\theta_{1} = {\theta_{2} = {\ldots = {\theta_{d - 2} = \frac{( {m - 1} )\pi}{M - 1}}}}} \\{\theta_{d - 1} = {2p\frac{( {m - 1} )\pi}{M - 1}}}\end{matrix} \},{m = {1\mspace{14mu} \ldots \mspace{11mu} M}},} & (6)\end{matrix}$

where pεN−{0} defines the density of the winding around the surface ofthe hypersphere. For p=1 the hypershpere will be sampled with only asingle traversal, while for higher numbers of p, denser windings can beachieved. Using (5) and (6) the unit vector sampling can be written as:

$\begin{matrix}{ \begin{matrix}{n_{m} =} & {\{ {x_{1}^{m},x_{2}^{m},\ldots \mspace{14mu},x_{d}^{m}} \}^{T} =} \\ = & \begin{Bmatrix}{{\cos \lbrack \frac{( {m - 1} )\pi}{M - 1} \rbrack},} \\{,{{\sin \lbrack \frac{( {m - 1} )\pi}{M - 1} \rbrack}{\cos \lbrack \frac{( {m - 1} )\pi}{M - 1} \rbrack}},} \\{,{{\sin^{2}\lbrack \frac{( {m - 1} )\pi}{M - 1} \rbrack}{\cos \lbrack \frac{( {m - 1} )\pi}{M - 1} \rbrack}},} \\\vdots \\{,{{\sin^{d - 2}\lbrack \frac{( {m - 1} )\pi}{M - 1} \rbrack}{\sin \lbrack {2p\frac{( {m - 1} )\pi}{M - 1}} \rbrack}}}\end{Bmatrix}^{T}\end{matrix} \}.} & (7)\end{matrix}$

An example sampling of a 3-sphere is shown in FIG. 1 for p=8 and M=64.

The Inverse Problem

The systemic view of both the forward and the inverse problems are shownin FIG. 2. In the case of the inverse problem, it is shown that itssolution produces the parameters of the fractal function provided thatfield distribution of the volume z(r) has been experimentally determinedand therefore is known.

More specifically, given an array of measurements z_(ijk) ^(e), i=1 . .. I, j=1 . . . J, k=1 . . . K over a region of size L^(d) one may thenidentify the parameters γ and φ_(mn) of a d-dimensional volume z(r) thatbest fits those measurements. In previous studies the identification ofthose parameters was done only for the 2 Dimensional surface. In orderto achieve the inverse identification of the volume parameters theapproach is similar to that described in Michopoulos et al.

The inner sum in Eq. 4 can be written as,

$\begin{matrix}{{{\sum\limits_{n = 0}^{N}{{\gamma^{n{({D - {({d + 1})}})}}( {1 - ^{\; \frac{2\pi}{L}\gamma^{n}{n_{m} \cdot r}}} )}^{\; \varphi_{mn}}}} = {{c_{m}(r)}^{T}\varphi_{m}}},} & (8)\end{matrix}$

where c_(m)(r) and φ_(m) are vectors in P^((N+1)) given by,

$\begin{matrix}{{{c_{m}(r)} = {\{ {c_{m\; 0},c_{m\; 1},\ldots \mspace{14mu},c_{mN}} \}^{T} = \begin{Bmatrix}{{\gamma^{{({D - {({d + 1})}})}0}( {1 - ^{{\frac{2\pi}{L}\gamma^{0}n_{m}}{\cdot r}}} )},} \\{{\gamma^{{({D - {({d + 1})}})}1}( {1 - ^{{\frac{2\pi}{L}\gamma^{1}n_{m}}{\cdot r}}} )},...} \\{\gamma^{{({D - {({d + 1})}})}N}( {1 - ^{{\frac{2\pi}{L}\gamma^{N}n_{m}}{\cdot r}}} )}\end{Bmatrix}^{T}}},} & (9) \\{\mspace{79mu} {{and},{\varphi_{m} = {\{ {^{{\varphi}_{m\; 0}},^{{\varphi}_{m\; 2}},\ldots \mspace{14mu},^{{\varphi}_{m\; N}}} \}^{T}.}}}} & (10)\end{matrix}$

By setting,

$\begin{matrix}{{{\,^{*}Q} = {{L( \frac{G}{L} )}^{D - 2}\sqrt{\frac{\ln \; \gamma}{M}}}},} & (11)\end{matrix}$

Eq. 4 can be written as:

$\begin{matrix}{{W(r)} = {{\,^{*}Q}{\sum\limits_{m = 1}^{M}{c_{m}^{T}{\varphi_{m}.}}}}} & (12)\end{matrix}$

By coalescing the vectors c_(m) and φ_(m) into larger vectors inP^(M(N+1)) such as,

c(r)={c ₁ ^(T) ,c ₂ ^(T) , . . . ,c _(M) ^(T)}^(T),  (13)

and

φ={φ₁ ^(T),φ₂ ^(T), . . . ,φ_(M) ^(T)}^(T),  (14)

Eq. 12 can be written as:

W(r)=*Qc(r)^(T)φ.  (15)

For the needs of the inverse problem it is assumed that a number ofmeasurements at points r_(t) ^(e) exist for a volume represented asz^(e)(r_(t) ^(e)), t=1 . . . T, T≧M(N+1). We seek to identify a volumethat is described by Eq. 15 and approximates the experimental pointsz^(e)(r_(t) ^(e)).

To solve this problem we first form the following linear system,

$\begin{matrix}{{\begin{bmatrix}{W( r_{1} )} \\{W( r_{2} )} \\\begin{matrix}\vdots \\{W( r_{T} )}\end{matrix}\end{bmatrix} = \begin{bmatrix}{z^{}( r_{1} )} \\{z^{}( r_{2} )} \\\begin{matrix}\vdots \\{z^{}( r_{T} )}\end{matrix}\end{bmatrix}},{or},} & (16) \\{{{{\,^{*}Q}\begin{bmatrix}{c( r_{1} )}^{T} \\{c( r_{2} )}^{T} \\\begin{matrix}\ldots \\{c( r_{T} )}^{T}\end{matrix}\end{bmatrix}}\begin{bmatrix}\varphi_{1} \\\varphi_{2} \\\ldots \\\varphi_{M}\end{bmatrix}} = {\begin{bmatrix}{z^{}( r_{1} )} \\{z^{}( r_{2} )} \\\begin{matrix}\vdots \\{z^{}( r_{T} )}\end{matrix}\end{bmatrix}.}} & (17)\end{matrix}$

By expanding the vectors in Eq. 16, the system can be written as,

$\begin{matrix}{{{{Cp} = z},{with},{C = {{\,^{*}Q}\begin{bmatrix}{{c_{11}( r_{1} )},{c_{12}( r_{1} )},\ldots \mspace{14mu},{c_{21}( r_{1} )},\ldots \mspace{14mu},{c_{MN}( r_{1} )}} \\{{c_{11}( r_{2} )},{c_{12}( r_{2} )},\ldots \mspace{14mu},{c_{21}( r_{2} )},\ldots \mspace{14mu},{c_{MN}( r_{2} )}} \\\vdots \\{{c_{11}( r_{T} )},{c_{12}( r_{T} )},\ldots \mspace{14mu},{c_{21}( r_{T} )},\ldots \mspace{14mu},{c_{MN}( r_{T} )}}\end{bmatrix}}},,{p = \lbrack {\varphi_{11},\varphi_{12},\ldots \mspace{14mu},\varphi_{1N},\varphi_{21},\ldots \mspace{14mu},\varphi_{MN}} \rbrack^{T}}}{{{and}\mspace{14mu} z} = {\begin{bmatrix}{z^{}( r_{1} )} \\{z^{}( r_{2} )} \\\begin{matrix}\vdots \\{z^{}( r_{T} )}\end{matrix}\end{bmatrix}.}}} & (18)\end{matrix}$

The system of Eq. 17 is an overdetermined system of M(N+1) equations.Since the right hand side vector z contains experimental measurements,it also contains noise; the system cannot, in general, have an exactsolution. Nevertheless, one can seek a p, such as Πp-zΠ is minimized,where Π_(p)Γ. is the vector norm. Such a p is known as the least squaressolution to the over-determined system. It should be noted that the lefthand side expression of Eq. 16 yields results in the complex domain, butas long as a minimal solution is achieved for real numbers on the righthand side, the imaginary parts will be close to 0. A solution can begiven by the following equation,

p=Vy,  (19)

where V is calculated by the Singular Value Decomposition (SVD) of C as,

UDV ^(T) =C,  (20)

where y is a vector defined as y_(t)=b_(t)′/d_(t), b=b_(t) is a vectorgiven by,

b=U ^(T) p),  (21)

and d_(i) is the i th entry of the diagonal of D.

The solution of the inverse problem as described by the overdeterminedsystem of Eq. 16 gives the phases φ_(mn) given known values of the otherparameters. In a general volume the only other parameter that is unknownis γ. As was pointed out in Michopoulos et al parameters G and D don'tneed to be considered as unknowns to be determined in this optimization.This is because for any combination of the phases it is always possibleto find new values for φ_(mn), that result in generating the samesurface.

Numerical Experiments

In order to assess the quality of the high dimensional fractalcharacterization results of the numerical examples that follow, wedefine the following error function,

$\begin{matrix}{{{error} = \frac{\sum\limits_{i = 1}^{P}{\frac{z_{i}^{} - z_{i}^{d}}{{\max ( z_{i} )}^{d} - {\min ( z_{i}^{d} )}}}}{P}},} & (22)\end{matrix}$

where z_(i) ^(e) is the value of the scalar field of the experimentallymeasured (or reference) points, P is the number of those points. In thefollowing examples P is set as P=I×J×K=40×40×40=64000 points. z_(i) ^(d)is the value of the scalar field of the inversely identified fractal andis equal to the real part of the truncated W-M function of Eq. (4)(z_(i) ^(d)=Re{W(r)}).

TABLE 1 PARAMETERS OF THE REFERENCE MODEL. Parameter Value L 1 G 1 × 10⁶γ 1.5 D 2.3 M 10 N 16 d 3

To study the feasibility of the invention, a few numerical experimentswere designed in the 3D space. The first experiment was based onsynthetic data and is aimed at inversely identifying only the phases ofa volume constructed by the fractal itself. The original volume is shownin FIG. 3 and was constructed using Eq. (4) with random phases and theparameters as shown in Table 1. The inversely identified volume usingthe phases resulting from Eq. (19) is presented in FIG. 3. It should benoted that this difference is very small compared to the magnitude ofthe volume and any discrepancies should be considered as the numericalerror of the SVD algorithm.

The second synthetic experiment involved the identification of both thephases and the γ parameter. An exhaustive search approach was adopted inthis case, as the sensitivity of the SVD inversion relative to the valueof γ is also of interest. For a range of the possible values forparameter γ the inversion of the phases was executed and the value ofthe error function (Eq. 22) was calculated. The error for various valuesof γ is presented on FIG. 4. The smallest value for the error is atγ=1.5, which is the one used originally for the generation of the volume(Table 1).

In order to investigate the behavior of the invention under noisy data,we also applied the inversion in the original data of FIG. 3, but forvarious levels of injected noise. The noise levels varied relative tothe range of the scalar field values from 0% to 300%. The results of theinverse identification for noise levels 20%, 40% and 300% are presentedin FIG. 5 and show impressively robust response. Even for noise levelsof 300% (FIG. 5), where the volume characteristics are visuallynon-existent, the fractal inversion methodology resembles (FIG. 5) theactual field very well. In FIG. 6 the mean absolute error of theidentification for various levels of noise is presented, where it can beobserved that as the noise levels increase, the mean absolute errorincreases linearly.

Although the previous analysis demonstrates the consistency andperformance of the invention, it is much more useful when applied toactual data. For this reason, a numerical test is performed based onmicro-tomography scans of a failed AS4/3501-6 composite specimen usedfor 3 degrees of freedom in-plane mechanical loading. AS4/3501-6 is acarbon/epoxy composite used in general purpose structural applicationsand the associated prepreg is manufactured by Hexcel Corporation. Thedata set used consisted of 243 tomographic slices of the specimen storedas digital images of resolution 1256×588 pixels each.

Because the data set is very large, the volume characterization was doneon a subset of the volumetric data shown in FIG. 7. The subset size was16×16×16 voxels and was taken from the region depicted by the red squarein FIG. 8. The fractal parameters were chosen to be: γ=1.5,D=2.3μL=1,G=1*10⁶ μM=80 and N=50. The inversion of the volume gives verysatisfactory results as shown in FIG. 8. For this test the mean absoluteerror as calculated by Eq. (22) was 4.74%.

To investigate the effect the choice of γ has on the fractal inversion,a parametric study was performed for various values of γ ranging from1.02 to 2. For each of those values the fractal phases where identifiedand the values of the mean absolute error from Eq. (22) were calculated.The results of this study are plotted in FIG. 9. Although the originalchoice of γ in the previous example resulted in a very satisfactoryinverse identification of the failed composite specimen volume, it canbe seen from FIG. 9, that even better results can be achieved for avalue of γ equal to 1.24. In that particular case the mean absoluteerror of the identified volume relative to the original was about 0.84%and the resulting volume is depicted in FIG. 7. This result underlinesthat for optimum characterization purposes on has to consider that theset of parameters to be identified should include both the phases and γ.

It should be noted that the present invention can be accomplished byexecuting one or more sequences of one or more computer-readableinstructions read into a memory of one or more computers from volatileor non-volatile computer-readable media capable of storing and/ortransferring computer programs or computer-readable instructions forexecution by one or more computers. Volatile computer readable mediathat can be used can include a compact disk, hard disk, floppy disk,tape, magneto-optical disk, PROM (EPROM, EEPROM, flash EPROM), DRAM,SRAM, SDRAM, or any other magnetic medium; punch card, paper tape, orany other physical medium. Non-volatile media can include a memory suchas a dynamic memory in a computer. In addition, computer readable mediathat can be used to store and/or transmit instructions for carrying outmethods described herein can include non-physical media such as anelectromagnetic carrier wave, acoustic wave, or light wave such as thosegenerated during radio wave and infrared data communications.

While specific embodiments of the present invention have been shown anddescribed, it should be understood that other modifications,substitutions and alternatives are apparent to one of ordinary skill inthe art. Such modifications, substitutions and alternatives can be madewithout departing from the spirit and scope of the invention, whichshould be determined from the appended claims.

What is claimed as new and desired to be protected by Letters Patent ofthe United States is:
 1. A computer implemented method for obtaining ananalytical representation of an internal structure and spatialproperties distribution of a selected physical domain, comprising:identifying d-dimensional correspondences of measured spatial propertiesor field distributions; and applying an inverse algorithm to thed-dimensional spatial properties or field distributions to calculate theWeierstrass-Mandelbrot (W-M) fractal model to thereby determineparameters defining an analytical and continuous Weierstrass-Mandelbrot(W-M) representation.
 2. The method of claim 1, wherein the physicaldomain is selected from a metal alloy, a carbon epoxy composite, or abiological tissue.
 3. The method of claim 1, wherein the physical domainis a man-made structure selected from a microprocessor, an aircraft, aship, an automobile, a darn, a road layer, a bridge, or a building. 4.The method of claim 1, wherein the physical domain is a live organism orpart of a live organism or a in vitro version of them (e.g wood).
 5. Themethod of claim 1, wherein the physical domain is a digitalrepresentation of a real or artificial object.
 6. The method of claim 5,wherein the digital representation is a spatial distribution of a scalarfield within a volume such as magnetic resonance slice images, x-raytomography data, ultrasonic data trough a volume, or any simulatedentity distributed in a volume.
 7. The method of claim 1, wherein thephysical domain is a landform selected from a mountain, a lake, part ofa landform, a planet, part of a planet, a solar system, part of a solarsystem, a cluster of solar systems, part of a cluster of solar systems,a galaxy, part of a galaxy, a cluster of galaxies, or part of a clusterof galaxies.
 8. The method of claim 1, wherein the d-dimensionalmodeling is obtained by applying a formula $\begin{matrix}{{{W(r)}=={\sqrt{\frac{\gamma}{M}}{\sum\limits_{m = 1}^{M}{\sum\limits_{n = {- \infty}}^{\infty}{{A_{m}( {1 - ^{\; k_{0}\gamma^{n}{n_{m} \cdot r}}} )}{^{\; \varphi_{mn}}( {k_{0}\gamma^{n}} )}^{D - {({d + 1})}}}}}}},} & (3)\end{matrix}$ where n_(m) are d-dimensional unit.
 9. The method of claim8, wherein based on an array of measurements over a selected region ofthe domain, parameters of the d-dimensional W-M fractal model areidentified that best fit the array of measurements by decomposing theproblem so as to apply singular value decomposition for determiningunknown phases of the W-M fractal model.
 10. A computer software productcomprising a physical computer-readable medium including storedinstructions that, when executed by a computer, cause the computer to:identify d-dimensional correspondences of measured spatial properties orfield distributions; and apply an inverse algorithm to the d-dimensionalspatial properties or field distributions to calculate theWeierstrass-Mandelbrot (W-M) fractal model to thereby determineparameters defining an analytical and continuous Weierstrass-Mandelbrot(W-M) representation.
 11. The computer software product of claim 10,wherein the physical domain is selected from a metal alloy, a carbonepoxy composite, or a biological tissue.
 12. The computer softwareproduct of claim 10, wherein the physical domain is a man-made structureselected from a microprocessor, an aircraft, a ship, an automobile, adam, a road layer, a bridge, or a building.
 13. The computer softwareproduct of claim 10, wherein the physical domain is a live organism orpart of a live organism or a in vitro version of them (e.g wood). 14.The computer software product of claim 10, wherein the physical domainis a digital representation of a real or artificial object.
 15. Thecomputer software product of claim 14, wherein the digitalrepresentation is a spatial distribution of a scalar field within avolume such as magnetic resonance slice images, x-ray tomography data,ultrasonic data trough a volume, or any simulated entity distributed ina volume.
 16. The computer software product of claim 10, wherein thephysical domain is a landform selected from a mountain, a lake, part ofa landform, a planet, part of a planet, a solar system, part of a solarsystem, a cluster of solar systems, part of a cluster of solar systems,a galaxy, part of a galaxy, a cluster of galaxies, or part of a clusterof galaxies.
 17. The computer software product of claim 10, wherein thed-dimensional modeling is obtained by applying a formula $\begin{matrix}{{{W(r)}=={\sqrt{\frac{\gamma}{M}}{\sum\limits_{m = 1}^{M}{\sum\limits_{n = {- \infty}}^{\infty}{{A_{m}( {1 - ^{\; k_{0}\gamma^{n}{n_{m} \cdot r}}} )}{^{\; \varphi_{mn}}( {k_{0}\gamma^{n}} )}^{D - {({d + 1})}}}}}}},} & (3)\end{matrix}$ where n_(m) are d-dimensional unit.
 18. The computersoftware product of claim 17, wherein based on an array of measurementsover a selected region of the domain, parameters of the d-dimensionalW-M fractal model are identified that best fit the array of measurementsby decomposing the problem so as to apply singular value decompositionfor determining unknown phases of the W-M fractal model.